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Isomorphism Testing Parameterized by Genus and Beyond

Daniel Neuen

TL;DR

The algorithm provides the first explicit upper bound on the dependence on g for an fpt isomorphism test parameterized by the Euler genus of the input graphs, and introduces $(t,k)-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm.

Abstract

We give an isomorphism test for graphs of Euler genus $g$ running in time $2^{O(g^4 \log g)}n^{O(1)}$. Our algorithm provides the first explicit upper bound on the dependence on $g$ for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time $f(g)n$ for some function $f$ (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude $K_{3,h}$ as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce $(t,k)$-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This concept may be of independent interest.

Isomorphism Testing Parameterized by Genus and Beyond

TL;DR

The algorithm provides the first explicit upper bound on the dependence on g for an fpt isomorphism test parameterized by the Euler genus of the input graphs, and introduces $(t,k)-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm.

Abstract

We give an isomorphism test for graphs of Euler genus running in time . Our algorithm provides the first explicit upper bound on the dependence on for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time for some function (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude as a minor. For such graphs, no fpt isomorphism test was known before. The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce -WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This concept may be of independent interest.

Paper Structure

This paper contains 15 sections, 15 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Weisfeiler-Leman Algorithm
  5. Group Theory
  6. Permutation Groups
  7. Algorithms for Permutation Groups
  8. Groups with Restricted Composition Factors
  9. Group-Theoretic Tools for Isomorphism Testing
  10. Allowing Weisfeiler and Leman to Split Small Color Classes
  11. Structure Theory and Small Color Classes
  12. Finding Disjoint and Connected Subgraphs
  13. A Decomposition Theorem
  14. An FPT Isomorphism Test for Graphs of Small Genus
  15. Conclusion

Key Result

Theorem 1.1

The Graph Isomorphism Problem for graphs excluding $K_{3,h}$ as a minor can be solved in time $2^{{\mathcal{O}}(h^4 \log h)}n^{{\mathcal{O}}(1)}$.

Figures (5)

  • Figure 1: Visualization of the construction of layer $(j,1)$ ($j$ even) of the graph $H_i$ in the proof of Theorem \ref{['thm:bounding-group-t-k-wl']}. Every vertex in $W_{j,1}^i$ is associated with a color $c$ in the image of $\chi_{j,1}^i$ and the corresponding color class $(\chi_{j,1}^i)^{-1}(c)$ shown next to the vertex. All color classes of size at most $t$ (the figure shows $t = 2$) containing only diagonal entries are split into singletons.
  • Figure 2: Visualization of the construction of layer $(j,r)$ ($j$ odd) of the graph $H_i$ in the proof of Theorem \ref{['thm:bounding-group-t-k-wl']}. Every vertex in $W_{j,r}^i$ is associated with a color $c$ in the image of $\chi_{j,r}^i$ and the corresponding color class $(\chi_{j,r}^i)^{-1}(c)$ shown next to the vertex. The edge colors $1$ and $2$ are depicted in orange and violet, respectively. For example, the left-most yellow vertex represents the color pair $(\chi_{j,r-1}^i(v_1,v_2), \chi_{j,r-1}^i(v_2,v_1)) = (\chi_{j,r-1}^i(v_2,v_1), \chi_{j,r-1}^i(v_1,v_2))$ and thus, the incident edge to the vertex representing $\chi_{j,r-1}^i(v_1,v_2) = \chi_{j,r-1}^i(v_2,v_1)$ (the left-most red vertex) has color $\{1,2\}$. Also, ${\mathcal{M}}_{r-1}(v_1,v_2) = \{\!\{(\chi_{j,r-1}^i(v_1,v_2), \chi_{j,r-1}^i(v_1,v_1)),(\chi_{j,-1}^i(v_2,v_2), \chi_{j,r-1}^i(v_1,v_2)),\dots\}\!\}$ and the two color-pairs in the multiset are represented by the second and fifth yellow vertex from the left.
  • Figure 3: Visualization for the construction of $H_1,H_2,H_3$ in Lemma \ref{['la:disjoint-trees']}. The sets $E(H_1),E(H_2),E(H_3)$ are marked in orange, violet and blue, respectively. Observe that the edge $(c_0,c_2) \in E(F)$ allows expansion which means the color class $c_2 \in V(F)$ is removed first in the inductive process. Afterwards, the leaves $c_3$ and $c_4$ are removed, and the color $c_1$ is added to the set $C_R$ since $G[V_{c_1},V_{c_3}]$ is isomorphic to $2K_{2,3}$.
  • Figure 4: The figure depicts the case $q \geq 3$ in the proof of Lemma \ref{['la:disjoint-trees']}. The partition ${\mathcal{A}}$ is shown in gray and parts of the sets $E(H_1),E(H_2),E(H_3)$ are marked in orange, violet and blue, respectively.
  • Figure 5: The figure depicts the case $q = 2$ and $|N_G(v) \cap V_c| = 1$ for all $v \in V_d$ in the proof of Lemma \ref{['la:disjoint-trees']}. The partition ${\mathcal{A}}^*$ consists of four blocks shown in gray. We need to contract the four equivalence classes, since otherwise it may happen that $v_1$ and $v_2$ appear in the same block where $\{v_r\} = H_r' \cap V_d$ and $H_r'$ is the subgraphs obtained from the induction hypothesis, $r \in \{1,2,3\}$. In this case, only one of the two subgraphs $H_1'$ and $H_2'$ could be extended to color $c$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 2.1: cf. Seress03
  • Definition 2.2
  • Theorem 2.3: Miller Miller83b
  • Theorem 2.4: Neuen Neuen22b
  • Definition 3.1
  • Theorem 3.2: Neuen22b
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 30 more